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/* cdf/betainv.c
*
* Copyright (C) 2004 Free Software Foundation, Inc.
* Copyright (C) 2006 Brian Gough
* Written by Jason H. Stover.
* Modified for GSL by Brian Gough
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
*/
/*
* Invert the Beta distribution.
*
* References:
*
* Roger W. Abernathy and Robert P. Smith. "Applying Series Expansion
* to the Inverse Beta Distribution to Find Percentiles of the
* F-Distribution," ACM Transactions on Mathematical Software, volume
* 19, number 4, December 1993, pages 474-480.
*
* G.W. Hill and A.W. Davis. "Generalized asymptotic expansions of a
* Cornish-Fisher type," Annals of Mathematical Statistics, volume 39,
* number 8, August 1968, pages 1264-1273.
*/
#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_cdf.h>
#include <gsl/gsl_randist.h>
#include "error.h"
double
gsl_cdf_beta_Pinv (const double P, const double a, const double b)
{
double x, mean;
if (P < 0.0 || P > 1.0)
{
CDF_ERROR ("P must be in range 0 < P < 1", GSL_EDOM);
}
if (a < 0.0)
{
CDF_ERROR ("a < 0", GSL_EDOM);
}
if (b < 0.0)
{
CDF_ERROR ("b < 0", GSL_EDOM);
}
if (P == 0.0)
{
return 0.0;
}
if (P == 1.0)
{
return 1.0;
}
if (P > 0.5)
{
return gsl_cdf_beta_Qinv (1 - P, a, b);
}
mean = a / (a + b);
if (P < 0.1)
{
/* small x */
double lg_ab = gsl_sf_lngamma (a + b);
double lg_a = gsl_sf_lngamma (a);
double lg_b = gsl_sf_lngamma (b);
double lx = (log (a) + lg_a + lg_b - lg_ab + log (P)) / a;
if (lx <= 0) {
x = exp (lx); /* first approximation */
x *= pow (1 - x, -(b - 1) / a); /* second approximation */
} else {
x = mean;
}
if (x > mean)
x = mean;
}
else
{
/* Use expected value as first guess */
x = mean;
}
{
double lambda, dP, phi;
unsigned int n = 0;
start:
dP = P - gsl_cdf_beta_P (x, a, b);
phi = gsl_ran_beta_pdf (x, a, b);
if (dP == 0.0 || n++ > 64)
goto end;
lambda = dP / GSL_MAX (2 * fabs (dP / x), phi);
{
double step0 = lambda;
double step1 = -((a - 1) / x - (b - 1) / (1 - x)) * lambda * lambda / 2;
double step = step0;
if (fabs (step1) < fabs (step0))
{
step += step1;
}
else
{
/* scale back step to a reasonable size when too large */
step *= 2 * fabs (step0 / step1);
};
if (x + step > 0 && x + step < 1)
{
x += step;
}
else
{
x = sqrt (x) * sqrt (mean); /* try a new starting point */
}
if (fabs (step0) > 1e-10 * x)
goto start;
}
end:
return x;
}
}
double
gsl_cdf_beta_Qinv (const double Q, const double a, const double b)
{
if (Q < 0.0 || Q > 1.0)
{
CDF_ERROR ("Q must be inside range 0 < Q < 1", GSL_EDOM);
}
if (a < 0.0)
{
CDF_ERROR ("a < 0", GSL_EDOM);
}
if (b < 0.0)
{
CDF_ERROR ("b < 0", GSL_EDOM);
}
if (Q == 0.0)
{
return 1.0;
}
if (Q == 1.0)
{
return 0.0;
}
if (Q > 0.5)
{
return gsl_cdf_beta_Pinv (1 - Q, a, b);
}
else
{
return 1 - gsl_cdf_beta_Pinv (Q, b, a);
};
}
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